Statistical tolerance analysis determines the probable or likely maximum varia-tion possible for a selected dimension. Similar to worst-case tolerance analysis, all tolerances and other variables are added to obtain the total variation. This method, however, more realistically assumes that it is highly improbable that all the dimensions in the tolerance stackup will be at their worst-case low limit or high limit at the same time. Remember, the worst-case tolerance stackup result requires some dimensions to be at their low limit and others to be at their high limit. So the direction of the deviation as well as the amount of deviation must be just so to achieve a worst-case condition.
It is more likely that the actual variation will be different than what is predicted by the worst-case model. In many cases, the sum of the dimensions and tolerances will likely approximate a normal distribution. Most or all of the dimensions will likely be closer to their nominal value than either extreme. Also, some of the dimen-sions that the worst-case model required to be at their upper limit may actually be closer to their lower limit, and vice versa. The combination of these factors leads to the idea of a statistical tolerance stackup. Generally, statistical tolerance analysis yields a smaller value for the total variation than a worst-case tolerance analysis performed on the same stackup. That is, statistical tolerance analysis techniques usually predict less variation than the worst-case results for a tolerance stackup.
This can be very beneficial from a functional point of view, as a lower overall pre-dicted variation will allow the design engineer the latitude to increase the tolerances allowed for manufacturing or design the fits between mating parts tighter, lead-ing to smaller gaps and higher perceived quality, or some combination of both. Of course, statistic tolerancing should only be used in cases where it is applicable.
A question arises as to when it is appropriate to use statistical versus worst-case tolerance analysis. The answer to this question depends on a number of fac-tors, including the number of tolerances in the tolerance stackup, the quantity of parts to be manufactured, manufacturing process controls, design sensitivity, past company practices, and willingness to accept risk, to name a few. A simple rule of thumb is as the number of tolerances in a tolerance stackup increases, the ben-efits and validity of using a statistical analysis increases. There are various rules in industry that state that for more than 3, 4, 6, 10, etc., dimensions a statistical analysis is the right choice.
This author does not adhere to the idea of an arbitrary number of dimensions being an automatic reason to switch from a worst-case to a statistical approach.
No doubt, as the number of tolerances in a tolerance stackup increases, a statistical solution not only may be a good idea, but may more accurately represent the
98 Mechanical Tolerance Stackup and Analysis, Second Edition
variation that will be seen at assembly. The number of tolerances alone, however, is insufficient reason to select a statistical approach. All factors, especially those factors relating to manufacturing and process controls, must be considered and weighed against the risk of an overly conservative or overly liberal result.
Statistical tolerance analyses are based on several conditions being in place.
These include
The manufacturing processes for the parts must be controlled processes.
•
This requires, among other things, that manufacturing nominal is the same as design nominal. (This is not always the case, however.)
Processes must be centered and output normal or Gaussian distributions.
•
(See Figure 8.1.) This presents a problem where unequal bilateral or uni-lateral tolerances have been specified. Six Sigma statistical tolerance analysis strategies address the tendencies of distributions to move off center or drift over time. Chapter 21 introduces the idea of mean shift as it relates to process centering.
Parts must be randomly selected for assembly.
•
This statement is based on the idea of interchangeability, from
mechani-•
cal engineering, and the idea of independence (or independent variables), from statistics.
Technically, for certain statistical tolerance analysis models, each variable
•
that contributes to the tolerance stackup must be independent from the other variables that affect the tolerance stackup. This requirement comes from statistics and requirements for random (or unrelated) variables. That is, each variable must be random and vary independently from the other variables in the tolerance stackup. This is often not the case with manufactured parts.
Consider a machined part with two tolerances (variables) that
contrib-•
ute to a tolerance stackup. It is possible that the tolerances are related, 0HDQ
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FIgure 8.1 Gaussian distribution.
Statistical Tolerance Analysis 99
perhaps from the associated features being machined in the same setup, or using one feature as the datum feature for the other. It is very likely that two features machined in the same setup will show similar trends in variation, both affected by the particular setup.
The same could be said for the tolerances of all features of a cast part or
•
all features produced by a common part of the die in a mold.
These variables are not truly independent, as they share common
influ-•
ences in the manufacturing process.
The design must be able to tolerate the possibility that some small
per-•
centage of the as-produced parts or assemblies exceed the calculated statistical result.
The enterprise must be willing to tolerate the possibility that some parts or
•
assemblies will be rejected due to exceeding the calculated statistical result.
There are several statistical methods available for tolerance analysis. Root-sum-square (RSS) and Monte Carlo simulations are the two most common.
Root-sum-square is commonly used on manually modeled and spreadsheet-based statistical tolerance stackups.
As presented in Chapter 3, design nominal and manufacturing nominal are rarely if ever the same. This presents a problem when considering the above assumptions. The requirements for the RSS statistical tolerance analysis methods used in this text are that manufacturing processes shall be centered and output normal distributions. This correlates to the Cp and Cpkvalues encountered in statis-tical process control (SPC), which address process spread and process centering.
It is beyond the scope of this text to address the statistical implications regarding these apparent discontinuities in great detail, but these topics are discussed a bit in Chapter 21. These topics are addressed in greater detail in Advanced Tolerance Stackup and Analysis by Bryan R. Fischer (2011). One solution to the difference between the design nominal and manufacturing nominal and the fact that some processes aren’t as controlled as they should be is to multiply the statistical result by a coefficient greater than 1. This practice also addresses the fact that most of the conditions listed in the previous paragraph are not always 100% applicable to every dimension and tolerance. Multiplying the RSS result by a coefficient greater than 1 gives the adjusted statistical result.
Monte Carlo simulation is typically used with computer-based tolerance analy-sis simulation software, but may also be used with spreadsheet models. Simply put, Monte Carlo simulations take all the variables in a tolerance stackup, assign each a random value within their range, derive a result, save the results, iterate this process thousands of times, average the results and possibly present predicted sta-tistical distributions. This is a purely stasta-tistical approach. As stated above, Monte Carlo analysis is often used with 3D tolerance analysis software tools; however, at least one 3D tolerance analysis software package uses more precise modeling algorithms. These are very powerful tools and are great for solving 3D tolerance stackups, as these tools allow the tolerance analyst to look at many combinations of translational and rotational variation. These tools are also fairly expensive and
100 Mechanical Tolerance Stackup and Analysis, Second Edition
may be complex to learn and use. 3D simulation tools are becoming easier to use with each release, but are still complex enough to warrant having dedicated staff to use them effectively. Refer to the information and case study in Chapter 21, which presents material on 3D tolerance analysis and includes a brief introduction to Six Sigma concepts and Sigmetrix’s CETOL 6 Sigma 3D analysis software. CETOL 6 Sigma is a powerful 3D tolerance analysis modeling tool that does not use Monte Carlo simulation. Thus, it solves for worst-case and statistical variation.
As presented in Chapter 7, worst-case tolerance analysis is used to calculate and predict the maximum variation possible, and the minimum and maximum limits of the subject of the tolerance stackup. With worst-case, the results strictly provide numerical information, vector values representing the variation and limits resulting from adding the variation to and subtracting it from the nominal value.
With worst-case tolerance analysis, all we are able to learn from the tolerance stackup is how much variation is possible and how that variation affects the sub-ject of the tolerance stackup. With statistical tolerance analysis, the same type of numerical information may be obtained. Statistical values for the probable toler-ance may be calculated, and similar to worst-case, these probable tolertoler-ances may be added to or subtracted from the nominal distance or angle to obtain statistical minimum and statistical maximum limits. As stated above, the statistical result is usually less than the worst-case result, so the statistical numbers will be different than the worst-case numbers. In this regard, statistical tolerance analysis is almost exactly the same as worst-case tolerance analysis, except the variation is not the maximum possible variation; it is the maximum probable variation that is likely to be encountered. The material that follows, which explains the RSS method, uses the statistical results and calculates the statistical minimum and maximum values for each tolerance stackup under consideration. However, there is another way to use statistical tolerance analyses.
Statistical tolerance analyses may also be used to obtain predictions of the number of defects that may be encountered (percent defects) for a population of parts and assemblies. The statistical tolerance analysis results may be set up to show how many parts or assemblies will fall within a certain range of variation, and by contrast, how many parts or assemblies will fall outside that range. The methods for performing these tolerance analyses are exactly the same as those that follow, except additional data is needed for the variables that contribute to the tolerance stackup. More statistical information is needed, specifically SPC data, and thus a better understanding of the manufacturing processes and their results is required. Such in-depth information and understanding is not always avail-able, so the statistical approach used here is simplified and intended to determine probable variation, not percent defects. Six Sigma methods require more sophisti-cated SPC and statistical data, such as standard deviations, Cp and Cpk values, and present variation, percent defects, and statistical distribution data in their results.
Chapter 21 provides a very brief introduction to Six Sigma concepts, and good examples of 3D tolerance analysis software.
Aside from Chapter 21, the statistical material in this text uses the RSS method for statistical solutions. This method takes each tolerance value, squares it, adds
Statistical Tolerance Analysis 101
the squared tolerance values, and takes the square root of the result, hence the name root-sum-square (RSS). The formula can be seen in Figure 8.2. This result is the RSS statistical tolerance. There are several variations on this method, using combinations of worst-case and statistical tolerancing, adjusting the result by mul-tiplying it by a value >1, or using standard deviations instead of tolerance values to obtain percent defects. Again, this text uses the RSS approach, and uses toler-ances rather than standard deviations. This method is more universally applica-ble, especially where statistical process control data is not available. For examples and a discussion of tolerance analysis using Six Sigma Strategies, expanded use of statistical data, and calculation of percent rejects, refer to Advanced Tolerance Stackup and Analysis by Bryan R. Fischer (2011). The reader is also directed to the Dimensioning and Tolerancing Handbook by Paul J. Drake, Jr. (1999, McGraw-Hill), which contains several chapters devoted to the study of various statistical tolerancing techniques.
I have been asked many times what the RSS tolerance stackup result rep-resents in terms of sigma (σ) or standard deviations. Students want to know if an RSS tolerance stackup represents a ±1σ, ±3σ, ±6σ, etc., distribution. It is generally assumed that if all the individual tolerances entered into the tolerance stackup are produced by processes controlled to ±3σ, then the RSS tolerance stackup result also represents ±3σ. To put it another way, it is generally assumed that the level of process controls of the inputs represents the level of process controls of the output (see Figure 8.3). Likewise if all the component tolerances are assumed to be ±1σ, ±2σ, or ±6σ, then the RSS tolerance stackup result rep-resents ±1σ, ±2σ, or ±6σ, respectively. The better you know your processes, the more accurate the statistical tolerance stackup result. It is very important to learn about the manufacturing processes where possible and to obtain reliable data from statistically controlled processes.
In practice, it is likely that the processes used to manufacture all the part fea-tures in a tolerance stackup and their associated tolerances are not controlled to the same level. That is, the tolerances in a tolerance stackup are probably manu-factured using a few ±2σ processes, a few ±3σ processes, a few ±4σ processes, etc., and perhaps even some processes where the level of control is unknown. So, in many environments it is likely that the tolerances in a tolerance stackup rep-resent a mixture of process capabilities. In some environments, especially where SPC is not practiced and process data are not collected, process capabilities for the individual tolerances are simply not known. This is especially true where other factors enter into the equation, such as datum feature shift or assembly shift. Assembly shift is particularly problematic, as it is a function of the assembly
RSS Tolerance = T12
+ T22
+ T32
+ ... Tn2
Where:
Tn = Tolerances in the Tolerance Stackup FIgure 8.2 Root-sum-square formula for statistical tolerancing.
102 Mechanical Tolerance Stackup and Analysis, Second Edition
process, and unless the assembly process is monitored and measured like every other process, it is likely not controlled. In manual assembly operations, assembly shift often manifests itself in its worst-case form. See Chapters 7 and 9 for more information on assembly shift. Again, this is all the more reason to use an adjust-ment factor when interpreting statistical results.
A step-by-step explanation of how to perform statistical tolerance stackups follows. This is exactly the same process as presented in Chapter 7, except for a few additional steps in which the tolerance values are squared and the square root of their sum is taken and multiplied by an adjustment factor as described above. These methods can be easily performed simultaneously using specialized spreadsheet software.
stAtIstIcAl tolerAnce stAckup WIth dImensIons Differences from worst-case are highlighted in italics.
1. Select the distance (gap or interference) whose variation is to be determined. Label one end of the distance A and the other end B (see Figure 8.4).
2. Determine if a one-, two-, or three-dimensional analysis is required.
a. If a two-dimensional analysis is required, determine if both direc-tions can be resolved into one dimension using trigonometry. If not, a linear tolerance stackup is not appropriate, and a computer pro-gram should be used for the tolerance analysis.
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Statistical Tolerance Analysis 103
b. If a three-dimensional analysis is required, a linear tolerance stackup is probably not appropriate, and a computer program should be used for the tolerance analysis. See Chapter 21 for more information about 3D tolerance analysis.
3. Determine a positive direction and a negative direction.
a. The positive direction in a tolerance stackup is easy to assign. The positive direction is the direction from point A to point B. Once the sides of the gap or distance being studied are labeled as A and B, the positive direction is the direction pointing from A toward B. (Note:
the method used to determine the positive and negative directions is defined differently in this edition of the text. The method defined here is simpler.)
b. Positive dimensions are indicated by placing a “+” sign adjacent to the dimension value (see Figure 8.5). Dimensions should also be assigned a direction by placing a dimension origin symbol at the end where the dimension starts and an arrowhead at the other end where the dimension terminates. All dimensions in the chain of dimensions and tolerances that are followed in the direction from A toward B should be labeled as positive dimensions. All dimensions that are followed in the opposite direction should be labeled as negative dimensions.
c. Now build the chain of dimensions and tolerances. Always start at Point A. If the direction of the dimension originating at A points toward B, then label it positive using a “+” sign, a dimension ori-gin symbol, and arrowhead as described in item 3.a above. If the dimension points away from B, label it negative using a “–” sign (see
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104 Mechanical Tolerance Stackup and Analysis, Second Edition
Figure 8.6). Identify the chain of dimensions and tolerances from point A to point B, and label all dimensions in the same direction positive or negative.
d. Follow the chain of dimensions and tolerances from point A to point B. You should be able to follow a continuous path from the start to the end of each dimension in the chain from point A to point B (see Figure 8.7). In this example, the first dimension starts at point A and ends at the left edge of the part. The second dimension starts where the first dimension ends, and ends at the right edge of the part.
The third dimension starts where the second dimension ends. The fourth dimension starts where the third dimension ends, and ends at point B. If the dimensions are not properly labeled, the nominal distance may be negative after the negative total is subtracted from the positive total. If this happens, check the + or – labels assigned to the dimensions, making sure that the sum of the positively labeled dimensions is larger than the sum of the negatively labeled dimen-sions. Remember that the total value of the positive dimensions must include distance A-B.
4. Convert all dimensions and tolerances to equal-bilateral format (± the same value; see Figure 8.8). Instructions for how to do this are included in Chapter 4.
5. Now all the dimensions and tolerances are entered into a chart and totaled for reporting purposes. Place each positive dimension value in the positive column on a separate line. Place each negative dimension value in the negative column on a separate line (see Figure 8.9.)
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Statistical Tolerance Analysis 105
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